Friday, February 19, 2021

Considering A Classic Abstract Algebra Problem

 Let (G, o) and (H, o) be groups. Then a homomorphism of (G, o) into (H, o) is a map of the sets G and H which has the following property:  f(x o y) = f(x) o f(y)


(G, o) = (R1 +)

(H, o) = (R*, ·)

Take f = the exponential function, so f(x) = exp (x), f(y) = exp(y)

Then: f(x + y) = exp(x + y) = exp(x) exp(y) = f(x) f(y)


H = R* = {x Î R: x  not equal 0}

And: exp R -> R* so exp(x + y) = exp(x) exp(y)

Def.: Isomorphism: An isomorphism of G onto H [(G, o), (H, o)] is a bijective homomorphism.

Example: H = P = {x Î R: x > 0}         (P, x)

Let G = (0, 1, 2, 3) for the operation (o) which is addition in Z4

Let H = (2, 4, 6, 8) for the operation (o) which is multiplication in Z10

Problem: Prepare the respective tables for the relevant isomorphism and give specific examples in terms of the function φ, i.e. show specific mappings.  (Where: φ(x) φ(y) = φ(xy) for example)

Hint: Check out this earlier blog post on isomorphisms: 

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